High School

The length of rose stems follows a normal distribution with a mean length of 19.78 inches and a standard deviation of 3.787 inches. A flower shop sells roses as parts of wedding flowers, wedding bouquets, and corsages. Please use this information to answer the following questions and use R (not the z-table) for any calculations.

a. What is the probability that a given rose stem will be shorter than 17.3 inches?
Answer: Round to at least FOUR digits after the decimal if necessary.

b. Suppose a rose is considered a 'long stem rose' if its stem length is longer than 23.8 inches. What is the probability that a given rose will be considered a long stem rose?
Answer: Round to at least FOUR digits after the decimal if necessary.

c. The flower shop has a rule that the shortest 8% of roses are clipped and used as corsages. What is the maximum stem length (in inches) a rose can be and still qualify to be used as a corsage by the shop?
Answer: inches. Round to at least FOUR digits after the decimal if necessary.

d. Suppose the z-score (standardized score) of a rose stem length is given as 1.26. Which of the following statements is a correct interpretation of the meaning of this value?
A. The length of this stem is 1.26 times longer than the average rose stem.
B. The length of this stem is 1.26 inches longer than the average rose stem.
C. The length of this stem is 1.26 standard deviations longer than the average rose stem.
D. There is not enough information provided to interpret this value.

Answer :

Final answer:

To find specific probabilities or values in a normal distribution, calculate the Z-score first. Use the pnorm function in R for probabilities and the qnorm function for values. A Z-score represents how many standard deviations a value is from the mean.

Explanation:

The length of rose stems can be analyzed using a Normal Distribution and the Z-Score concept.

  1. To find the probability that a given rose stem will be shorter than 17.3 inches, we need to calculate the Z-score which is (17.3 - 19.78) / 3.787 to obtain the Z value. This value can then be plugged into R's pnorm function to get the probability.

  2. For a 'long stem rose' longer than 23.8 inches, the Z-score will be (23.8 - 19.78) / 3.787. The pnorm function in R will then give 1 - the probability because we want longer than this length, not shorter.

  3. Using R's qnorm function with 0.08 (for the shortest 8%) as the probability, the mean, and the standard deviation as arguments, we can find the maximum stem length a rose can be to qualify as a corsage.(qnrom(0.08, 19.78, 3.787))

  4. A Z-score of 1.26 means that the length of the stem is 1.26 standard deviations longer than the average rose stem. So, answer C is correct.

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